Course syllabus
Flerdimensionell analys
Calculus in Several Variables
FMAB30, 6 credits, G1 (First Cycle)
Valid for: 2019/20
Decided by: PLED F/Pi
Date of Decision: 2019-03-26
General Information
Main field: Technology.
Compulsory for: B2, BI2, BME2, C2, D2, E1, F1, I1, K2, L2, M1, MD1, N2, V2, W2
Elective for: IBYA3, IBYI3, IBYV3, IDA3, IEA3
Language of instruction: The course will be given in Swedish
Aim
The course aims at giving a basic treatment of calculus in
several variables. Particular emphasis is given to the role this
plays in applications in different subjects of technology, in order
to give the future engineer a good foundation for further studies
in mathematics as well as other subjects. The aim is furthermore to
develop the student's ability to solve problems and to assimilate
mathematical text.
Learning outcomes
Knowledge and understanding
For a passing grade the student must
- be able to compute with and handle elementary functions
of several variables within the framework of the course with
confidence, together with their derivatives and integrals.
- be familiar with different representations of curves, surfaces
and volumes in two and three dimensions, and be able to use them in
computations.
- be able to carry out (specified) changes of variables in
partial differential equations, and by this means to solve such
equations.
- be familiar with the basic theory of optimization, local as
well as global, and be able to find the solution in simple
cases.
- be able to demonstrate an ability to independently choose
methods to evaluate double and triple integrals, and be able to
carry out the solution essentially correctly.
- be able to demonstrate an ability to independently choose
method to evaluate a curve integral, and be able to carry
out the solution essentially correctly.
- be able to demonstrate a good ability to carry out algebraic
calculations within the context of the course.
- be able to give a general account of and to illustrate the
meaning of such mathematical concepts in calculus in several
variables that are used to construct and study mathematical models
in the applications.
- be able to account for the contents of some central
definitions, theorems and proofs.
Competences and skills
For a passing grade the student must
- in the context of problem solving be able to demonstrate an
ability to independently choose and use mathematical concepts and
methods within calculus in several variables.
- in the context of problem solving be able to integrate concepts
from different parts of the course.
- be able to demonstrate an ability to construct and analyse some
simple mathematical models in calculus in several variables.
- be able to show capability to explain mathematical arguments in
a structured and logically clear way.
- have a basic ability to use Maple for visualisation and formula
manipulation, and be able to identify some of its possibilities and
limitations.
Contents
- Generalities on functions of several variables: level curves,
function surfaces, level surfaces, surfaces in parameter form,
curvilinear coordinates.
- Partial derivatives. Differentiability, tangent planes, error
propagation. The chain rule. Applications to partial differential
equations. Gradient, directional derivative, level curves. Study of
stationary points. Curves, tangents, arc length. Surfaces, normal
direction, tangent plane. Functional (Jacobi) matrix and
determinant, linearisation. Implicitly given functions.
- Optimization on compact and non-compact domains. Optimization
with constraints.
- Double and triple integrals. Iterated integration.
Change of variables. Improper integrals. Applications: volume,
centre of gravity.
- Curve integrals in the plane. Green's formula with
applications. Potentials and exact differentials.
- Computer work. Visualisation and formula manipulation using
Maple.
Examination details
Grading scale: TH - (U,3,4,5) - (Fail, Three, Four, Five)
Assessment: Written test comprising theory and problem solving. Computer work.
The examiner, in consultation with Disability Support Services, may deviate from the regular form of examination in order to provide a permanently disabled student with a form of examination equivalent to that of a student without a disability.
Parts
Code: 0117. Name: Calculus in Several Variables.
Credits: 6. Grading scale: TH.
Code: 0217. Name: Computer Work.
Credits: 0. Grading scale: UG.
Admission
Required prior knowledge: Single variable calculus, e g one of the courses FMAA01, FMAA05 or FMAA50, and linear algebra, e g one of the courses FMAB20, FMAA55 or FMAA20.
The number of participants is limited to: No
The course overlaps following course/s: FMA025, FMA435, FMA430, FMAB35
Reading list
- Jonas Månsson och Patrik Nordbeck: Flerdimensionell analys. Studentlitteratur, 2013, ISBN: 9789144080833.
- Övningar i flerdimensionell analys. Studentlitteratur, 2013, ISBN: 9789144092508.
Contact and other information
Course coordinator: Studierektor Anders Holst, Studierektor@math.lth.se
Course administrator: Studerandeexpeditionen, expedition@math.lth.se
Course homepage: http://www.maths.lth.se/course/flerdimnykod/